1. ENE 428 Microwave Engineerin g Lecture 10 Signal Flow Graphs and Excitation of Waveguides 1

2. Review (1) Two-port network - At low frequencies, the z, y, h, or ABCD parameters are basic network input- output parameter relations. The parameters are readily measured using short- and open-circuit tests at the terminals. - At RF or microwave frequency, these parameter are difficult to measure - At high frequencies (in microwave range), scattering parameters (S parameters) are defined in terms of traveling waves and completely characterize the behavior of two-port networks. - S parameters can be easily measured using matched loads which ensure the stability of the network. 2

3. Signal flow graphs and applications A convenient technique to represent and analyzed the transmission and reflection of waves in a microwave amplifier. Relations between the variables can be obtained using Mason’s rule. The flow graph technique permits expression, such as power gains and voltage gains of complex microwave amplifiers, to be derived easily. 3

4. Rules of signal flow graph constructions 1.Each variable is designated as a node. 2.The S parameters and reflection coefficients are represented by branches. 3.Branches enter dependent variable nodes and emanate from independent variable nodes. The independent variable nodes are the incident waves, and the reflected waves are dependent variable nodes. 4.A node is equal to the sum of the branches entering it. 4

5. Signal flow graph of the S parameters of a two-port network Observe that b 1 and b 2 are the dependent variable nodes and a 1 and a 2 are the independent variable nodes. 5

6. Signal flow graph of a signal generator 6

7. Signal flow graph of a load impedance 7

8. Signal flow graph of a microwave amplifier Observe that the nodes b g, a g, b L, and a L are identical to a 1, b 1, a 2, and b 2, respectively. 8

9. Mason’s rule Mason’s rule is used to determine the ratio of transfer function T of a dependent to an independent variable. 9

10. Variables’descriptions (1) P 1, P 2, (and so on) = paths connecting the dependent and independent variables whose transfer function T is to be determined. A path is defined as a set of consecutive, codirectional branches along which no node is encountered more than once as we move in the graph from the independent to the dependent node.  L(1) = the sum of all first-order loops. A first-order loop is defined as the product of the branches encountered in a round trip as we move from a node in the direction of the arrows back to that original node. 10

11. Variables’descriptions (2)  L(2) = the sum of all second-order loops. A second-order loop is defined as the product of any two nontouching first-order loops.  L(3) = the sum of all third-order loops. A third-order loop is defined as the product of any three nontouching first-order loops.  L(4),  L(5), and so on represent fourth-, fifth-, and higher order loops. 11

12. Variables’descriptions (3)  L(1) (P) = the sum of all first-order loops that do not touch the path P between the independent and dependent variables.  L(2) (P) = the sum of all second-order loops that do not touch the path P between the independent and dependent variables.  L(3) (P),  L(4) (P) and so on represent third-, fourth-, and higher order loops that do not touch the path P. 12

13. Ex1 Use Mason’s rule to obtain b 1 /b S as shown in a microwave amplifier’s signal flow graph 13

14. Applications of Signal flow graphs (1) The calculation of the input reflection coefficient,  IN Observing that P 1 = S 11, P 2 = S 21  L S 12,  L(1) = S 22  L, and  L(1) (1) = S 22  L, we can use Mason’s rule to obtain 14

15. Applications of Signal flow graphs (2) The calculation of the output reflection coefficient,  OUT Observing that P 1 = S 22, P 2 = S 21  S S 12,  L(1) = S 11  S, and  L(1) (1) = S 11  S, we can use Mason’s rule to obtain 15

16. Excitation of Waveguides Several propagating modes can be excited in the waveguide along with evanescent modes. Formalism for excitation of the give wg mode due to an arbitrary electric and magnetic current source will be determined. Excitation of the wg using aperture coupling will also be briefly discussed. 16

17. Single mode excitation using current sheets (TE case) 17

18. The amplitudes must be equal to satisfy BCs. 18 From eqn (5), at Z = 0 (a.k.a the XY plane): Similarly: (multiply by -1 on both sides, we’ll get:)

19. 19 When (5) is applied to (1) and (2), we get The discontinuity in the transverse magnetic field in (6) is equal to the electric surface current density thus at z = 0, therefore

20. The surface current density at z = 0 can be found. 20 This current will excite only the TE mn mode since Maxwell’s equations and all boundary conditions are satisfied.

21. The electric current that excites only the TM mode can be determined analogously. 21

22. and represent the transverse electric and magnetic field components while and are the longitudinal electric and magnetic field components. Excitation of WG modes by an arbitrary electric or magnetic current source. 22 where n represents any possible TE or TM mode.

23. The unknown amplitude A n + can be determined using the Lorentz reciprocity theorem. (1) 23  Let the volume V be the region between the wg walls,  Let and,  And let be the nth wg mode in the –z direction, where S is a closed surface enclosing the volume V, and are the fields due to the current source. (11)

24. The unknown amplitude A n + can be determined using the Lorentz reciprocity theorem. (2) 24 Substitution (12) and (13) into (11) with and, gives The portion of the surface integral over the wg walls vanishes because the tangential electric field is zero. This reduces the integration to the guide cross section, S 0, at the planes z 1 and z 2. (14) (15)

25. 25 The unknown amplitude A n + can be determined using the Lorentz reciprocity theorem. (3) results in Since (from EIE 208) for m  n always Therefore, in order for to give a nonzero value, m must be equal to n ( in other words, m = n). So, and must be of the same mode.

26. 26 From eqn (16), since the second term is zero, we’ll only have the 1 st term Using eqns (7) - (10) and (15), then reduces (14) to (16) On surface Z1: and On surface Z2: and Z1 Z2

27. The unknown amplitude A n + can be determined using the Lorentz reciprocity theorem. (4) 27 is a normalization const. proportional to the power flow of the n th mode Since the second integral vanishes, this further reduces to where In other words,

28. The unknown amplitude A n - can be determined using the same procedure. 28 By repeating with and, we get or, in other words These above results can be applied to any type of wg such as stripline and microstrip, where modal fields can be defined.

29. 29 A similar derivation can be carried out for a magnetic current source. This source will also generate positively and negatively traveling waves which can be expressed as a superposition of waveguide modes. For, the Lorentz Reciprocity theorem reduces to By following the same procedure as for the electric current case, the excitation coefficients of the nth waveguide mode can be derived as where

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